Page 254 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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46 Fluid Dynamics (Chemical Engineering)

within this subject is the subset of phenomena associated tational momentum, (4) balance of energy, (5) conserva-
specifically with the kinematic and dynamic behavior of tion of charge–current, (6) conservation of magnetic flux,
fluids. Kinematics is the study of motion per se, while and (7) thermodynamic irreversibility.
dynamics includes the response of specific materials to In the vast majority of situations of importance to chem-
applied forces. This requires one to apply the theory of ical engineers, the conservation of charge–current and
deformable continuum fields. In its most general form magnetic flux are of no importance, and therefore, we will
the continuum field theory includes both fluid mechanics not consider them further here. They would be of consid-
and dynamics in all their myriad forms. This article deals erable importance in a magnetohydrodynamic problem.
specifically with kinematic and dynamic applications. The four balance or conservation principles can all be
represented in terms of a general equation of balance writ-
ten in integral form as
I. INTRODUCTION
∂ψ
dV =− ψv · n ds − j Dψ · n ds
∂t
The phenomena of fluid mechanics are myriad and multi- V S S
form. In the practice of chemical engineering, most appli- Net increase Net convective Net diffusive
of ψ in V influx of ψ influx of ψ
cations of fluid mechanics are associated with either flow
through a bounded duct or flow around a fixed object in
+ ˙ r ψ dV (1)
the context of design of processing equipment. The de-
tails of such problems may be very simple or extremely V
Net production
complex. The chemical engineer must know how to ap- of ψ inV
ply standard theoretical and empirical procedures to solve
or in differential form as (n is the outward-directed normal
these problems. In cases where standard methods fail, he
vector; hence, − ψv · n ds represents influx)
or she must also know how to apply fundamental princi-
ples and develop an appropriate solution. To this end this ∂ψ
article deals with both the fundamentals and the applica- =−∇ · ψv −∇ · j Dψ + ˙ r ψ
∂t (2)
tion thereof to bounded duct flows and flows about objects Net increase Net convective Net diffusive Net production
of incompressible liquids of the type commonly encoun- of ψ at point influx of ψ influx of ψ of ψ at point
tered by practicing chemical engineers. The phenomena
where ψ represents the concentration or density of any
associated with compressible flow, two-phase gas–liquid
transportable property of any tensorial order, j Dψ repre-
flow, and flow through porous media are not considered
sents the diffusive transport flux of property ψ, and ˙ r ψ
because of space limitations.
represents the volumetric rate of production or generation
of property ψ within the volume V , which is bounded by
the surface S.
II. BASIC FIELD EQUATIONS Equation (2) is expressed in the Eulerian frame of ref-
(DIFFERENTIAL OR MICROSCOPIC) erence, in which the volume element under consideration
is fixed in space, and material is allowed to flow in and
A. Generic Principle of Balance out of the element. An equivalent representation of very
different appearance is the Lagrangian frame of reference,
The fundamental theory of fluid mechanics is expressed
in which the volume element under consideration moves
in the mathematical language of continuum tensor field
with the fluid and encapsulates a fixed mass of material so
calculus. An exhaustive treatment of this subject is found
that no flow of mass in or out is permitted. In this frame
in the treatise by Truesdell and Toupin (1960). Two fun-
of reference, Eq. (2) becomes
damental classes of equations are required: (1) the generic
equations of balance and (2) the constitutive relations. Dψ/Dt =−ψ∇ · v − ∇ · j Dψ + ˙ r ψ , (3)
Thegenericequationsofbalancearestatementsoftruth,
which is a priori self-evident and which must apply to all where the new differential term Dψ/Dt is called the sub-
continuum materials regardless of their individual char- stantial or material derivative of ψ and is defined by the
acteristics. Constitutive relations relate diffusive flux vec- relation
tors to concentration gradients through phenomenologi-
Dψ ∂ψ
cal parameters called transport coefficients. They describe = + v · ∇ψ. (4)
Dt ∂t
the detailed response characteristics of specific materials.
There are seven generic principles: (1) conservation of Equations (2) and (3) are related by an obvious vector
mass, (2) balance of linear momentum, (3) balance of ro- identity.